Transport in heterogeneous porous formations: Spatial moments, ergodicity, and effective dispersion

TLDR: In this paper, the first and second spatial moments are regarded as random functions of time, and their expected value and variance are derived in terms of the velocity field, and the moments are assumed to satisfy the ergodic hypothesis if their coefficients of variation are negligible.

Abstract: Transport of inert solutes in natural porous formations is dominated by convection and by the large-scale heterogeneity of permeability. A solute body inserted in the formation spreads because of the variation of velocity among and along the stream tubes which cross the plume. With neglect of the slow effect of pore-scale dispersion the solute particles preserve their initial concentration, but the body as a whole spreads in an irregular manner (Figures 1, 2, and ). The transport theory, based on representation of permeability and velocity as random space functions, can predict the expected value and variance of concentration, but under the above conditions, the coefficient of variation may be large. In contrast, the spatial moments of the solute body are less susceptible to uncertainty, depending on the transverse dimensions of the plume and on the travel time. The first and second spatial moments are regarded as random functions of time, and their expected value and variance are derived in terms of the velocity field. The moments are assumed to satisfy the ergodic hypothesis if their coefficients of variation are negligible. The conditions which ensure the fulfillment of this requirement are examined. The “effective dispersion coefficients” are defined with the aid of the spatial moments and are shown to depend generally on the initial size of the solute body and on travel time. The results are illustrated by an analytical solution of transport in a stratified formation with the average velocity parallel to the bedding.